Mathematical Control and Related Fields (MCRF)

Exact controllability for the Lamé system

Pages: 743 - 760, Volume 5, Issue 4, December 2015      doi:10.3934/mcrf.2015.5.743

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Belhassen Dehman - Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092 El Manar, Tunisia (email)
Jean-Pierre Raymond - Institut de Mathématiques de Toulouse, Université Paul Sabatier & CNRS, 31062 Toulouse Cedex, France (email)

Abstract: In this article, we prove an exact boundary controllability result for the isotropic elastic wave system in a bounded domain $\Omega$ of $\mathbb{R}^{3}$. This result is obtained under a microlocal condition linking the bicharacteristic paths of the system and the region of the boundary on which the control acts. This condition is to be compared with the so-called Geometric Control Condition by Bardos, Lebeau and Rauch [3]. The proof relies on microlocal tools, namely the propagation of the $C^{\infty}$ wave front and microlocal defect measures.

Keywords:  Lamé system, elasticity, controllability, geometric control condition, microlocal defect measures, propagation of wave front.
Mathematics Subject Classification:  Primary: 93C20, 93B05; Secondary: 74B05, 35L51, 35L05.

Received: August 2014;      Revised: May 2015;      Available Online: October 2015.